The room is empty except for two large chests. On each is carved a message strangely in English :. Back in the days of yore, five small towns decided they wanted to build roads directly connecting each pair of towns. While the towns had plenty of money to build roads as long and as winding as they wished, it was very important that the roads not intersect with each other as stop signs had not yet been invented. Also, tunnels and bridges were not allowed. Is it possible for each of these towns to build a road to each of the four other towns without creating any intersections?
One reason it is difficult to define discrete math is that it is a very broad description which encapsulates a large number of subjects.
In this course we will study four main topics: combinatorics the theory of ways things combine ; in particular, how to count these ways , sequences , symbolic logic , and graph theory. However, there are other topics that belong under the discrete umbrella, including computer science, abstract algebra, number theory, game theory, probability, and geometry some of these, particularly the last two, have both discrete and non-discrete variants. Ultimately the best way to learn what discrete math is about is to do it.
Let's get started! Before we can begin answering more complicated and fun problems, we must lay down some foundation. We start by reviewing mathematical statements, sets, and functions in the framework of discrete mathematics. Section 0. Adjective : Individually separate and distinct. Synonyms : separate - detached - distinct - abstract. Probability can be defined as identifying the possibility of occurrence of an event, in terms of mathematics, it is the detailed description of random processes and their relevant outcomes.
In order to represent the likelihood of an event, it is being shown in number between 0 and 1 included. The several laws of probability hold deep applicability in the variety of realms such as genetics, weather forecasting , stock markets, etc.
Besides that,. The most fundamental type of probability is uniform probability. Also, if the outcomes under a set are equally probable, then the probability of each event is equivalent to the ratio of cardinalities.
The product, sum and complement laws of probability act analogously as the same laws from combinatorics. Also, the structure of the principle of inclusion and exclusion PIE of probability is also the same as the PIE for sets.
Related article: An Introduction to Probability Distribution. Being a branch of mathematics, Set Theory is involved in collecting of objects. Sets can either be discrete or continuous; and at an initial level, set theory is implicated on why and how these sets can be organized, integrated, and counted. It includes;. The cardinality of a finite set is particularly the number of elements in that set. For a given set A, its cardinality can be expressed as A.
A complement of a set is the set of elements that are not included in that set. Also, the study of set complements provides the various number of methods in order to calculate cardinalities of finite sets. The union and intersection proffer several means for explaining how any combination of sets can be consolidated. The principle of inclusion and exclusion PIE contributes processes for identifying either the union or intersection between two or more sets. Suggested blog: What is Descriptive Analysis?
Boolean algebra describes the operations defined on variables that consider the value of true 1 or false 0. This is implemented to design computer or digital circuits via logic gates that accept signals as an input and gives signals as an output. A graph which has no cycle is called an acyclic graph. A tree is an acyclic graph or graph having no cycles. A tree is simply an acyclic graph or a graph having no cycle, and a general tree is the non-empty finite set of components known as vertices or nodes that hold the property that each and every node could embrace a minimum degree as 1 and maximum degree as n.
If specifying the binary tree then, in the conditions where the outdegree of each node is less than or equivalent to 2 under a directed tree, then the tree is known as the binary tree. And a tree that involves nodes such as an empty tree is a binary tree.
Some of the basic terminology and definitions under binary trees are;. Root: A binary tree has a single node known as a root of the tree. Left Child: The left node of the root is termed as its left child.
Right Child: The right node of the root is named as its right child. Parent: The parent node is the one that has a left child or right child or both. Siblings: Two nodes of the tree having the same parent are known as siblings. Leaf: A node that has no child is termed as a leaf. However, the number of leaves over a tree can be varied from a minimum as one to the maximum as half the number of vertices in a tree.
While concluding the blog, it can be said that discrete mathematics is the branch of mathematics that deals with various sets of objects taking into account only distinct, separated values. What you do is trial and error, experimentation, guesswork. We have seen how these topics branches under discrete mathematics are considered as significant factors within computer science,. Also check: Conditional Probability. For example, recursive algorithms rely on solutions provided by recursive relations, the design of digital circuits demand the knowledge of Boolean algebra, number theory is used for messaging systems and cryptography, etc.
Be a part of our Instagram community. Table of Content 1. A Function assigns to each element of a set, exactly one element of a related set. Functions find their application in various fields like representation of the computational complexity of algorithms, counting objects, study of sequences and strings, to name a few.
The third and final chapter of this part highlights the important aspects of functions. Since f is both surjective and injective , we can say f is bijective. The rules of mathematical logic specify methods of reasoning mathematical statements. Greek philosopher, Aristotle, was the pioneer of logical reasoning. Logical reasoning provides the theoretical base for many areas of mathematics and consequently computer science.
It has many practical applications in computer science like design of computing machines, artificial intelligence, definition of data structures for programming languages etc. The purpose is to analyze these statements either individually or in a composite manner. A propositional consists of propositional variables and connectives. We denote the propositional variables by capital letters A, B, etc.
The connectives connect the propositional variables. It is because unless we give a specific value of A, we cannot say whether the statement is true or false. It is false if A is true and B is false. The rest cases are true.
A Contradiction is a formula which is always false for every value of its propositional variables. A Contingency is a formula which has both some true and some false values for every value of its propositional variables. Duality principle states that for any true statement, the dual statement obtained by interchanging unions into intersections and vice versa and interchanging Universal set into Null set and vice versa is also true.
If dual of any statement is the statement itself, it is said self-dual statement. A compound statement is in conjunctive normal form if it is obtained by operating AND among variables negation of variables included connected with ORs. In terms of set operations, it is a compound statement obtained by Intersection among variables connected with Unions.
A compound statement is in disjunctive normal form if it is obtained by operating OR among variables negation of variables included connected with ANDs. In terms of set operations, it is a compound statement obtained by Union among variables connected with Intersections.
A predicate is an expression of one or more variables defined on some specific domain. A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable. The variable of predicates is quantified by quantifiers. Universal quantifier states that the statements within its scope are true for every value of the specific variable. Existential quantifier states that the statements within its scope are true for some values of the specific variable.
If we use a quantifier that appears within the scope of another quantifier, it is called nested quantifier. To deduce new statements from the statements whose truth that we already know, Rules of Inference are used. Mathematical logic is often used for logical proofs.
Proofs are valid arguments that determine the truth values of mathematical statements. An argument is a sequence of statements. The last statement is the conclusion and all its preceding statements are called premises or hypothesis. A valid argument is one where the conclusion follows from the truth values of the premises. Rules of Inference provide the templates or guidelines for constructing valid arguments from the statements that we already have.
Group Theory is a branch of mathematics and abstract algebra that defines an algebraic structure named as group. Generally, a group comprises of a set of elements and an operation over any two elements on that set to form a third element also in that set. These symbols are not in general convertible [commutative], but are associative. In this chapter, we will know about operators and postulates that form the basics of set theory, group theory and Boolean algebra. Any set of elements in a mathematical system may be defined with a set of operators and a number of postulates.
A binary operator defined on a set of elements is a rule that assigns to each pair of elements a unique element from that set. The postulates of a mathematical system form the basic assumptions from which rules can be deduced. A set is closed with respect to a binary operator if for every pair of elements in the set, the operator finds a unique element from that set. The set of positive integers excluding zero with addition operation is a semigroup.
A monoid is a semigroup with an identity element. An identity element is also called a unit element. The set of positive integers excluding zero with multiplication operation is a monoid.
Here identity element is 1. A group is a monoid with an inverse element. So, a group holds four properties simultaneously - i Closure, ii Associative, iii Identity element, iv Inverse element. The order of a group G is the number of elements in G and the order of an element in a group is the least positive integer n such that an is the identity element of that group G.
As all the matrices are non-singular they all have inverse elements which are also nonsingular matrices. Hence, inverse property also holds. So, a group holds five properties simultaneously - i Closure, ii Associative, iii Identity element, iv Inverse element, v Commutative. The set of positive integers including zero with addition operation is an abelian group. Here, identity element is 1. A cyclic group is a group that can be generated by a single element.
Every element of a cyclic group is a power of some specific element which is called a generator.
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